Adaptive cruise control method based on improved grey prediction

Abstract

In order to improve the comprehensive performance of adaptive cruise control system in the car-following process and take the safety into account, an improved model predictive control algorithm considering multi-performance objective optimization is designed. In the prediction model part, the grey Verhulst model with saturation state is introduced to predict the acceleration disturbance of the preceding vehicle, and the particle swarm optimization algorithm is used to estimate the parameters, which is then applied to the car following model. The control problem is transformed into a quadratic programming problem with multiple constraints through multi-objective quadratic performance index, and the vector constraint management method is introduced to solve the problem of no feasible solution caused by hard constraints. The emergency acceleration, deceleration and stable following are simulated. Finally, the Worldwide Harmonized Light Vehicles Test Cycle is co-simulated. The results show that the improved model predictive control algorithm can improve the tracking capability, fuel economy and comfort of adaptive cruise system.

Keywords

Adaptive cruise multi-performance objective optimization model predictive control grey prediction

1 Introduction

As an advanced driver assistance system (ADAS), adaptive cruise control (ACC) system determines the expected longitudinal acceleration of the vehicle and automatically adjusts the safe inter-vehicle distance according to the driving environment detected by on-board sensors such as camera, millimeter wave radar and lidar, It has the advantages of improving safety, ride comfort and traffic flow [1]. At present, most ACC systems adopt hierarchical control structure. The upper controller acts as the decision-making mechanism and the lower controller acts as the actuator [2].

Adaptive cruise control is divided into cruise control and following cruise. Traditional control methods such as PID control [3], fuzzy logic [4] and LQR control [5] have been widely used. Model predictive control (MPC) has become the more mainstream control algorithm in ACC because it can solve the contradictory control objectives by weighting [6]. In the design of traditional MPC prediction model, the acceleration disturbance of the preceding vehicle in the prediction horizon is simplified, and it is considered that the acceleration data of the preceding vehicle in the prediction horizon is the same as the current sampling time [7]. In practice, however, acceleration is not constant as the preceding vehicle accelerates and decelerates, which will lead to the optimization result of the rolling optimization is not optimal, thus reducing the comprehensive performance of ACC system. In order to improve the prediction accuracy of MPC algorithm, aiming at the problem of system disturbance, Luo [8] proposed to introduce the acceleration disturbance of preceding vehicle, which improved the prospective and anti-interference capability of distance control. Within the framework of model predictive control, Chao et al. [9] applied Markov chain based on neural network to estimate the vehicle speed, but data is huge and computational. Dai et al. [10] proposed a linear least squares estimator, which improves the fuel economy of ACC system and reduces the peak jerk. Wu et al. [11] combined the traditional grey prediction model GM (1,1) and model predictive control algorithm to improve the tracking capability and safety of ACC system under the condition of emergency acceleration and deceleration. Considering the disturbance of preceding vehicle acceleration, Li et al. [12] proposed an on-line prediction method of preceding vehicle acceleration based on Gaussian process, which improved the comfort of ACC system.

Based on the above prediction models, grey prediction is a method to predict the system with uncertain factors. Compared with other prediction algorithms, it is best to deal with the prediction objects with small samples and has high real-time performance. Considering that when the MPC algorithm has a large prediction horizon, the linear least squares (LS) prediction and the traditional Grey Prediction GM (1,1) model mainly solve the prediction problem of monotonic change sequence. The prediction ability of nonlinear data is not strong and is only suitable for short-term prediction. In this paper, grey Verhulst model (GVM) prediction [13], which is more suitable for car following model, is used to improve the prediction model link. The basic idea of Verhulst model [14] is that the data grows exponentially at the beginning, then the growth rate slows down gradually, and finally stabilizes at a fixed value. It is mainly used to describe the process with saturation state, that is, the “s” process. In order to improve the prediction accuracy and practical application of Verhulst model, Ding et al. [15] corrected the background value of the model to make the optimized model have the minimum error. He et al. [16] used the linear programming method to estimate the parameters in Verhulst model and solved it by genetic algorithm to improve the prediction accuracy. Deng et al. [17] used unbiased model to eliminate the self-deviation of grey model, and further improved the prediction accuracy by collecting the latest two data to compensate the error.

In order to improve the adaptability of the prediction model and solve the problem of large error in the solution result when the acceleration sequence fluctuates violently, the particle swarm optimization algorithm is used to estimate the parameters of GVM prediction. The fitness function is designed to make full use of the latest data and the saturation function is introduced to make the predicted value of acceleration more reasonable. The penalty function and vector constraint management method are used to solve the problem of no feasible solution caused by algorithm hard constraints. Finally, the ACC longitudinal motion control problem is transformed into an on-line quadratic programming problem.

2 Establishment of model predictive control algorithm

2.1 Longitudinal kinematic model

As shown in Fig. 1, the longitudinal kinematic relationship between ACC vehicle and target vehicle is given, and the definition is as follows.

Fig. 1

Longitudinal kinematics diagram of ACC system.

{\begin{matrix} Δ d = d - d_{des} \\ Δ v = v_{p} - v_{f} \end{matrix}

(1)

where Δd is the inter-vehicle distance error; Δv is the relative velocity of the car and the car in front; d and d_des are the actual and expected inter-vehicle distance, respectively; v_f and v_p are the velocity of the velocity of the ACC vehicle and the preceding vehicle, respectively.

In this paper, we adopted a constant time headway strategy[18], $d_{des} = t_{h} v_{f} + d_{0}$ (2) where t_h is the vehicle time headway, d₀ is minimum safe distance.

The relationship between the actual acceleration and the expected acceleration can be considered as a first-order inertial link, $a_{f} = \frac{K_{L}}{T_{L} s + 1} a_{des}$ (3)

Where a_f is the acceleration of ACC vehicle, a_des is the expected acceleration, K_Lis the system gain and T_L is the time constant.

According to the longitudinal kinematic relationship between the ACC vehicle and the preceding vehicle, the state vector is defined as x(k) = [Δd(k) , Δv(k) , a_f(k)] ^T, the longitudinal kinematics state equation is obtained as follows. $x (k + 1 | k) = Ax (k) + Bu (k) + Gw (k)$ (4) where u(k) is the control variable, u(k) = a_des(k),w(k) = a_p(k) is the acceleration perturbation of the target vehicle, and the corresponding coefficient matrix is as follows. $\begin{matrix} A = [\begin{matrix} 1 & T_{s} & - t_{h} T_{s} - \frac{1}{2} T_{s}^{2} \\ 0 & 1 & - T_{s} \\ 0 & 0 & 1 - \frac{T_{s}}{τ} \end{matrix}] \\ B = [\begin{matrix} 0 \\ 0 \\ \frac{T_{s} K_{L}}{τ} \end{matrix}] G = [\begin{matrix} \frac{1}{2} T_{s}^{2} \\ T_{s} \\ 0 \end{matrix}] \end{matrix}$ where T_s is the sampling period, τ is the delay constant of the lower controller, and u is the system control input.

Safety, tracking capability, comfort and fuel economy are important indexes for evaluating ACC vehicles. y(k) = [d(k) , Δv(k) , a(k)] ^T is defined as system output, i.e., $y (k) = Cx (k)$ (5)

Corresponding coefficient matrix: $C = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

P is the prediction horizon, N is the control time domain, P ⩾ N, assuming the current moment is k, the prediction horizon is [k, k + P - 1], To improve the accuracy of the car-following prediction model and the robustness of the MPC algorithm, an error correction term is introduced. formula (4) and formula (5) Gradually iterate and we can obtain $\begin{matrix} {\begin{matrix} x (k + 1 | k) = Ax (k) + Bu (k) + Gw (k) + λ e (k) \\ y (k) = Cx (k) \end{matrix} \\ {\begin{matrix} X = A_{p} x (k) + B_{p} U + G_{p} W_{p} + M_{p} e (k) \\ Y = C_{p} X \end{matrix} \end{matrix}$ (6) whereX, U, and W_p are the state sequence, control sequence and acceleration disturbance sequence respectively, Y is the output sequence, λ = diag {λ₁, λ₂, λ₃}, $C_{p} = diag {\begin{matrix} C & C & \dots & C \end{matrix}}_{p \times p}$ is the corresponding coefficient matrix. $e (k) = x (k) - x (k | k + 1)$ $M_{p} = [\begin{matrix} λ & A λ & . . . & A^{P - 1} \end{matrix} λ]^{T}$ $A_{p} = {[\begin{matrix} A^{1} \\ A^{2} \\ ⋮ \\ A^{P} \end{matrix}]}_{P \times 1}$ $B_{P} = {[\begin{matrix} B & 0 & \dots & 0 \\ AB & B & ⋱ & ⋮ \\ ⋮ & ⋮ & ⋱ & 0 \\ A^{P - 1} & A^{P - 2} & \dots & \sum_{i = 0}^{P - N} A^{i} B \end{matrix}]}_{P \times N}$ $G_{P} = {[\begin{matrix} G & 0 & \dots & 0 \\ AG & G & ⋱ & ⋮ \\ ⋮ & ⋮ & ⋱ & 0 \\ A^{P - 1} G & A^{P - 2} G & \dots & G \end{matrix}]}_{P \times P}$ $W_{p} = [\begin{matrix} a_{p} (k) & a_{p} (k + 1) & \dots & a_{p} (k + p - 1) \end{matrix}]_{p \times 1}^{T}$

2.2 Improvement of model predictive control algorithm

For the traditional MPC algorithm of the ACC longitudinal control model, the acceleration disturbance sequence of the front vehicle is set as the same at every moment in the prediction time domain, i.e., a_p(k) = a_p(k + 1) = a_p(k + P - 1), the acceleration of the vehicle is not constant during actual acceleration and deceleration driving, which will lead to the control sequence obtained by the rolling optimization link not being optimal, thus affecting the comprehensive performance of the ACC system.

2.2.1 Acceleration prediction based on Grey Verhulst model

Assume that the original acceleration sequence is: $X^{(0)} = {w^{(0)} (1), w^{(0)} (2), \dots, w^{(0)} (n)}$ (7)X⁽¹⁾ is the first-order generated sequence by accumulated generating operation $X^{(1)} = {w^{(1)} (1), w^{(1)} (2), \dots, w^{(1)} (n)}$ (8) where $X^{(1)} = \sum_{i = 1}^{k} w^{(1)} (n) 1 pt 1 pt 1 pt 1 pt 1 pt 1 pt (k = 1, 2, \dots, n)$ .

Z⁽¹⁾ is the sequence of mean generation of consecutive neighbors from X⁽¹⁾ $Z^{(1)} = {Z^{(1)} (1), Z^{(1)} (2), \dots, Z^{(1)} (n)}$ (9) where $Z^{(1)} = \frac{1}{2} [w^{(1)} (k) + w^{(1)} (k + 1)] 1 pt 1 pt 1 pt 1 pt 1 pt 1 pt 1 pt (k = 1, 2, \dots, n)$ , the grey Verhulst model and its whitenization differential equation are $\begin{matrix} X^{(0)} + {aZ}^{(1)} = b {(Z^{(1)})}^{2} \\ \frac{{dx}^{(1)}}{dt} + {ax}^{(1)} = b {(x^{(1)})}^{2} \end{matrix}$ (10)

Time response function is defined as $\hat{x} (k + 1) = \frac{{ax}^{(1)} (1)}{{bx}^{(1)} (1) + [a - {bx}^{(1)} (1)] e^{ak}}$ (11)

Estimates of parameters in definition formula can be obtained as $\hat{α} = {[a b]}^{T} = {(B^{T} B)}^{- 1} B^{T} Y$ (12)

here, $B = [\begin{matrix} - z^{(1)} (2) & {(z^{(1)} (2))}^{2} \\ - z^{(1)} (3) & {(z^{(1)} (3))}^{2} \\ ⋮ & ⋮ \\ - z^{(1)} (n) & {(z^{(1)} (n))}^{2} \end{matrix}], Y = [\begin{matrix} x^{(0)} (2) \\ x^{(0)} (3) \\ ⋮ \\ x^{(0)} (n) \end{matrix}]$

The predicted value of the original sequence is $x^{\hat{(0)}} (k + 1) = x^{\hat{(0)}} (k + 1) - x^{\hat{(0)}} (k)$ (13)

2.2.2 Parameter estimation based on particle swarm optimization

Through the study of traditional GVM, it is found that the new data of actual acceleration sequence are more closely related to future prediction, older data contributes less to future projections, especially the first value is not cumulative and is not regular. Therefore, in order to reduce the impact of historical data on prediction results and make full use of new data, improve the fitness function, the weight matrix is introduced to increase the importance of new data, and the fitness function is $fitness = \frac{1}{n} • \sum_{i = 1}^{n} | σ_{i} (X_{i}^{(1)} - {\hat{X}}_{i}^{(1)}) / X_{i}^{(1)} |$ (14) where $X_{i}^{(1)}$ and ${\hat{X}}_{i}^{(1)}$ are the cumulative value of the ith acceleration and the corresponding predicted value, respectively.

Considering that the actual acceleration and deceleration of vehicles are bounded, the saturation function is introduced to ensure the rationality of the predicted values. ${\overset{\land}{X}}_{i}^{(1)} = 1 pt 1 pt {\begin{matrix} a_{max} 1 pt & {\overset{\land}{X}}_{i}^{(1)} ⩾ a_{max} \\ {\overset{\land}{X}}_{i}^{(1)} 1 pt 1 pt & a_{min} ⩽ {\overset{\land}{X}}_{i}^{(1)} ⩽ a_{max} \\ a_{min} 1 pt 1 pt & {\overset{\land}{X}}_{i}^{(1)} ⩽ a_{min} \end{matrix}$ (15) wherea_max and a_min are the upper and lower bounds of the acceleration predictions, respectively.

Particle swarm optimization algorithm consists of the following four basic steps.

Initialize random particle swarm (i = 1, 2, ⋯ , N)

X_i =(x_i1, x_i2, …, x_id) , V_i =(v_i1, v_i2, …, v_id) are the d-dimensional position vector and velocity vector of the i-th particle in the swarm respectively, N is the number of particle swarm.

T he personal best and global best

The optimum value so far sought by the i-th particle is the personal best pbest =(p_i1, p_i2, …, p_id).

Select the minimum from the pbest and compare with historical extremes to choose the minimum as the current global best gbest =(p_g1, p_g2, …, p_gd).

Velocity and position iteration formulas $\begin{matrix} v_{id} = w * v_{id} + c_{1} r_{1} (p_{id} - x_{id}) + c_{2} r_{2} (p_{gd} - x_{gd}) \\ x_{id} = x_{id} + v_{id} \end{matrix}$ (16) where w is an inertia weight,c₁, c₂are acceleration factors, and r_i, r₂ ∈ [0, 1] are random variables.

The termination condition (maximum number of iterations or deviation between adjacent generations) is reached.

The improved model predictive control (IMPC) algorithm process is as follows.

Step 1: n(n ⩾ 4) continuous historical acceleration data are selected as the original data sequence of prediction, and the first-order cumulative sequence X⁽¹⁾ is generated.

Step 2: Initialize random particle swarm NPSO.

Step 3: Based on the values of a, b, the first-order cumulative estimates are calculated by Equation (11), and the predicted values of the original data sequence are calculated by Equation (13).

Step 4: Calculate the fitness value of each particle according to the fitness function (14).

Step 5: Compare the fitness value of the ith particle with its local best pbest, if smaller, then updatepbest, compare the fitness value with gbest, if smaller, then update gbest.

Step 6: Update the velocity and position of each particle according to Equation (16).

Step 7: If the termination condition is satisfied, the predicted value of the original data is output, otherwise returns to step 3.

Step 8: Update the acceleration disturbance sequence W_p according to the output predicted value and apply it to Equation (6) for rolling optimization.

Step 9: Update the original data sequence according to the acceleration information collected at the next sampling time and return to step 3.

Step 10: Continue the calculation of the predicted value at the next sampling time and update the value of W_p until the simulation time is reached.

2.3 ACC control target analysis

With the mass production and popularization of ACC, the index to measure the quality of ACC is no longer just a single performance index, but also takes into account the fuel economy and ride comfort under the premise of ensuring safety. However, these optimization objectives have constraints and conflicts. In order to make the ACC system take into account the above multiple optimization objectives in the following process, the quadratic performance index and linear inequality constraints are used to convert the upper-level expected acceleration problem into a constrained quadratic programming problem.

2.3.1 Tracking capability

The tracking capability of ACC vehicle is evaluated by distance error and relative velocity [19]. $J_{T} = w_{Δ d} Δ d^{2} + w_{Δ v} Δ v^{2}$ (17) where w_Δdand w_Δv are weight coefficients.

2.3.2 Fuel economy

According to the quantitative analysis of fuel economy [3], the fluctuation range of acceleration and acceleration change rate affects the fuel economy index. $J_{F} = w_{a} a_{des}^{2} + w_{j} {jerk}^{2}$ (18) wherew_aand w_j are weight coefficients.

2.3.3 Ride comfort

The ride comfort is related to the inter-vehicle distance error, the acceleration and jerk, and the driver reference acceleration is introduced [20]. $\begin{matrix} J_{C} = w_{Δ d} Δ d^{2} + (w_{a} a_{f, des}^{2} + w_{j} {jerk}^{2}) \\ + w_{c} {(a_{f, ref} - a_{f})}^{2} \\ a_{f, ref} = k_{v} Δ v + k_{d} Δ d \end{matrix}$ (19) where w_c,k_v and k_d are weight coefficients.

2.3.4 Safety

In order to avoid collision, hard constraints are applied to the vehicle distance. $d ⩾ d_{safe} = \max {t_{TTC} Δ v, d_{0}}$ (20) wheret_TTCis collision time, D is actual distance, d_safe is safe distance, and d₀ is minimum safe vehicle distance.

In summary, the linear inequality constraints are as follows. ${\begin{matrix} Δ d_{min} ⩽ Δ d ⩽ Δ d_{max} \\ Δ v_{min} ⩽ Δ v ⩽ Δ v_{max} \\ a_{f, min} ⩽ a_{f} ⩽ a_{f, \max} \\ {jerk}_{\min} ⩽ jerk ⩽ {jerk}_{max} \\ a_{f, \min} ⩽ a_{des} ⩽ a_{f, \max} \end{matrix}$

The hard constraints are relaxed by relaxation factors and relaxation coefficients to expand the feasible region, so as to solve the problem that the MPC algorithm adopts hard constraints and has no feasible solution in the rolling optimization process. ${\begin{matrix} u_{min} + ɛ_{1} v_{min}^{u} ⩽ u ⩽ u_{max} + ɛ_{1} v_{max}^{u} \\ {jerk}_{min} + ɛ_{2} v_{min}^{j} ⩽ jerk ⩽ {jerk}_{max} + ɛ_{2} v_{max}^{j} \\ y_{min} + ɛ_{3} v_{min}^{y} ⩽ y ⩽ y_{max} + ɛ_{3} v_{max}^{y} \end{matrix}$ (21) whereɛ_i ⩾ 0(i = 1, 2, 3) is relaxation factor, $v_{max}^{u}, v_{max}^{j}, v_{max}^{y}, v_{min}^{u}, v_{min}^{j}, v_{min}^{y}$ are the upper and lower relaxation coefficients of hard constraints, respectively.

A quadratic penalty term is added to avoid the constraint inequality losing its restriction due to the infinite increase of relaxation factor, Finally, the performance cost function with predictive horizon [k + 1, k + P] and control horizon [k, k + N - 1] is established as follows. $\begin{matrix} J = \sum_{i = 1}^{P} (J_{T} (k + i) + J_{c} (k + i)) \\ + \sum_{i = 1}^{N} J_{F} (k + i) + ɛ^{T} ρ ɛ \\ = Y_{p}^{T} {WY}_{P} + U^{T} RU + ɛ^{T} ρ ɛ \end{matrix}$ (22) here, Y_p and U are the performance index vector and the matrix of control variables in time domain P, W, R, and ρ are weight coefficients, and ɛis Relaxation factor vector.

The simultaneous (17)–(21) converts the rolling optimization into an online quadratic programming problem with constraints, i.e., ${\begin{matrix} \min {U^{T} HU + 2 FU} \\ s . t Ω U ⩽ T \end{matrix}$ (23) here,Ωand T are constraint matrix. $U = [\begin{matrix} U \\ ɛ \end{matrix}] H = [\begin{matrix} B_{P}^{T} C_{P}^{T} {WC}_{P} B_{P} + R & 0 \\ 0 & ρ \end{matrix}]$ $\begin{matrix} F = & [(x {(k)}^{T} A_{P}^{T} C_{P}^{T} + W_{P}^{T} G_{P}^{T} C_{P}^{T} + e (k) M_{p}^{T}) \\ {{WB}_{P} 0]}^{T} \end{matrix}$

3 Numerical simulation

The numerical simulation analysis is carried out by MATLAB, and the IMPC algorithm is compared with the ordinary model predictive control (OMPC) algorithm, the traditional grey prediction improved algorithm (MPC_GM(1,1)) and the least squares prediction improved algorithm (MPC_LS). The parameter setting is shown in Table 1.

Table 1
Simulation parameters

Parameter Value Parameter Value

P 10 T _s 0.2

N 4 τ 0.4

d ₀ 5 t _h 1.5

$V_{max}^{Y}$ [0.1;1;0.1] k _v 0.25

$V_{min}^{Y}$ [–0.1;–1;–0.1] k _d 0.02

$v_{max}^{u}$ 0.1 $v_{max}^{j}$ 0.5

$v_{min}^{u}$ –0.1 $v_{min}^{j}$ –0.5

σ (1,2,3,4) ρ (3,3,3)

w _a 1 w _d 1

w _j 0.006 w _v 1

M 40 w _c 0.1

Parameter	Value	Parameter	Value
P	10	T _s	0.2
N	4	τ	0.4
d ₀	5	t _h	1.5
$V_{max}^{Y}$	[0.1;1;0.1]	k _v	0.25
$V_{min}^{Y}$	[–0.1;–1;–0.1]	k _d	0.02
$v_{max}^{u}$	0.1	$v_{max}^{j}$	0.5
$v_{min}^{u}$	–0.1	$v_{min}^{j}$	–0.5
σ	(1,2,3,4)	ρ	(3,3,3)
w _a	1	w _d	1
w _j	0.006	w _v	1
M	40	w _c	0.1

3.1 Emergency deceleration and acceleration condition

In case of the condition of emergency deceleration and acceleration, it is assumed that the ACC vehicle and the preceding vehicle are driving with the same initial velocity, both are 20 m/s, The acceleration of the preceding vehicle decreases linearly in the first 10 seconds to –1 m/ s², at the time t = 10–16 s, the preceding vehicle accelerates with an acceleration 3m/s², then decelerates with -2m/s² at t = 25 - 30s. The simulation results are shown in Fig. 2.

Fig. 2

Simulation comparison of emergency deceleration and acceleration condition (a). Distance errorΔd. (b). Velocity v_f. (c). Acceleration a_f.

It can be seen from Fig. 2 that compared with other prediction model algorithms, the IMPC algorithm has better prediction effect. The vehicle velocity response is faster when the vehicle changes from deceleration to acceleration in the 10th second, and it also converges fastest when the preceding vehicle velocity tends to be stable from rapid deceleration in the 30th second. In the deceleration phase of the 25th second, the distance error of IMPC algorithm always maintains positive value, which ensures the safety under the condition of rapid deceleration, and has less fluctuation than other algorithms, and converges faster to 0. In the acceleration diagram, the acceleration curve and the overall fluctuation is more stable, and the acceleration amplitude is minimum. Finally, there is no obvious overshoot in the acceleration from rapid deceleration to uniform velocity, and the fastest stabilized, which is better comfort for the driver. In summary, IMPC improves the ability of vehicles to coordinate multiple targets in emergency situations.

The tracking index is defined as follows [19]. $eTEI = \frac{1}{N} \sum_{k = 1}^{N} (ξ_{DV} | Δ d | + | Δ v (k) |)$ (24) where ξ_DV is the weight coefficient, taken as 0.2.

The fuel economy is calculated according to the EMIT consumption model [21]; During driving, the smaller the jerk, the higher the comfort [22], the comfort is calculated by means of the average jerk. The smaller the value, the higher the comfort.

Quantify the performance indicators and draw bar diagrams as shown in Fig. 3. Compared with OMPC, IMPC improves 15.2%, 5.9%, and 10.3% in terms of tracking capability, fuel economy and comfort under the emergency condition.

Fig. 3

Performance index analysis bar chart of emergency deceleration and acceleration condition.

3.2 Stable following condition

In case of the condition of stable following, it is assumed that the ACC vehicle and the preceding vehicle are driving with the same initial velocity, both are 35 m/s, In the 10th s, the preceding vehicle decelerates slowly with an acceleration of –0.6 m/s² and then drives at a constant velocity. In the 20th s, the preceding vehicle decelerates slowly again with an acceleration of –0.8 m/s² and then maintains a constant velocity. The simulation results are shown in Fig. 4.

Fig. 4

Simulation comparison of stable following condition. (a). Distance errorΔd. (b). Velocity v_f. (c). Acceleration a_f.

As shown in Fig. 4, compared with other prediction model algorithms, IMPC algorithm has smaller vehicle distance error and converges to 0 fastest, which improves the dynamic tracking capability. The velocity response is the fastest and tends to be stable first when the preceding vehicle decelerates slowly twice and then drives at a constant speed. The acceleration fluctuation is smaller without overshoot, and converges to 0 fastest, ensuring the ride comfort.

Quantify the performance indicators and draw the bar chart Fig. 5. Under this stable following condition, the IMPC algorithm achieves an 11.8% improvement in tracking performance, 11.7% improvement in fuel economy and 23.1% improvement in comfort over the OMPC.

Fig. 5

Performance index analysis bar chart of stable following condition.

4 Co-simulation

Longitudinal motion control model, inverse longitudinal dynamic model and vehicle dynamic model controlled by IMPC algorithm are established by MATLAB/Simulink, and co-simulated with Prescan as shown in Fig. 6. Some simulation parameters are set according to Table 1.

Fig. 6

Co-simulation model of ACC system.

In order to simulate the real road conditions, the simulation verification is carried out under Worldwide Harmonized Light Vehicles Test Cycle (WLTC) conditions [23]. the IMPC algorithm is compared with OMPC, MPC_GM(1,1) and MPC_LS by quantifying performance indicators, Verify the correctness and validity of the algorithm. The condition of WLTC is more complex, with the highest speed of 131 km/h. It pays more attention to the instantaneous working conditions and hardly drives uniformly. It can accurately simulate the urban and highways conditions. The simulation results are shown in Fig. 7.

Fig. 7

Co-simulation comparison of WLTC condition. (a). Distance errorΔd. (b). Velocity v_f. (c). Acceleration a_f.

As shown in Fig. 7 (a), the fluctuation range of the distance error curve is the smallest under the control of IMPC algorithm, which improves the dynamic tracking capability of ACC vehicles. The local amplification diagrams of (b) and (c) show that the velocity and acceleration fluctuations are the smallest, the response is the fastest and the convergence is greatly improved, thus improving the comfort and fuel economy.

Through the quantitative calculation of each performance index and drawing the histogram as shown in Fig. 8, the IMPC algorithm can be obtained to improve the tracking capability by 16.8 %, the fuel economy by 2.86 %, and the comfort by 8.9 % on the basis of OMPC.

Fig. 8

Performance index analysis bar chart of WLTC condition.

5 Conclusion

Based on the MPC control algorithm, by introducing the preceding vehicle acceleration disturbance estimator, an improved GVM prediction is proposed to improve the performance of MPC. A fitness function that makes full use of new data is designed, and the particle swarm optimization algorithm is used to estimate the parameters of GVM prediction. Through numerical simulation comparison, it is verified that the model has better adaptability to acceleration sequence, and the IMPC algorithm improves the comprehensive performance of ACC system under emergency acceleration and deceleration conditions and soothing conditions. The WLTC condition is co-simulated to simulate the driving conditions of real cities, suburbs and highways. The results show that compared with the OMPC, MPC_GM(1,1) and MPC_GLS algorithms, the IMPC algorithm proposed in this paper improves the dynamic tracking capability, fuel economy and comfort of the ACC system under the premise of ensuring traffic safety.

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 61701397 and Grant 51705419, and supported by the Fundamental Research Funds for the Central Universities, CHD (Grant no. 300102211513, 300102210512, 300102210511, 300102211514, 300102211521).

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Adaptive cruise control method based on improved grey prediction

Abstract

Keywords

1 Introduction

2 Establishment of model predictive control algorithm

2.1 Longitudinal kinematic model

2.2.1 Acceleration prediction based on Grey Verhulst model

2.3.1 Tracking capability

Table 1 Simulation parameters Parameter Value Parameter Value P 10 T s 0.2 N 4 τ 0.4 d 0 5 t h 1.5 V max Y [0.1;1;0.1] k v 0.25 V min Y [–0.1;–1;–0.1] k d 0.02 v max u 0.1 v max j 0.5 v min u –0.1 v min j –0.5 σ (1,2,3,4) ρ (3,3,3) w a 1 w d 1 w j 0.006 w v 1 M 40 w c 0.1

Funding

References